L8a4

L8a3

L8a5

	Visit L8a4's page at Knotilus! Visit L8a4's page at the original Knot Atlas!
	L8a4 is $8_{12}^{2}$ in the Rolfsen table of links.

L8a4 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X10,4,11,3 X12,10,13,9 X16,13,5,14 X14,7,15,8 X8,15,9,16 X2536 X4,12,1,11
Gauss code	{1, -7, 2, -8}, {7, -1, 5, -6, 3, -2, 8, -3, 4, -5, 6, -4}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)^{3}}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{5/2}-3q^{3/2}+4{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {5}{q^{3/2}}}-{\frac {6}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {2}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{5}-a^{5}z^{-1}+2z^{3}a^{3}+4za^{3}+3a^{3}z^{-1}-z^{5}a-3z^{3}a-4za-2az^{-1}+z^{3}a^{-1}+za^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{7}-az^{7}-2a^{4}z^{6}-5a^{2}z^{6}-3z^{6}-2a^{5}z^{5}-4a^{3}z^{5}-5az^{5}-3z^{5}a^{-1}-a^{6}z^{4}+5a^{2}z^{4}-z^{4}a^{-2}+3z^{4}+3a^{5}z^{3}+9a^{3}z^{3}+11az^{3}+5z^{3}a^{-1}+2a^{6}z^{2}+4a^{4}z^{2}+2a^{2}z^{2}+z^{2}a^{-2}+z^{2}-2a^{5}z-7a^{3}z-7az-2za^{-1}-a^{6}-3a^{4}-3a^{2}+a^{5}z^{-1}+3a^{3}z^{-1}+2az^{-1}$ (db)

Vassiliev invariants

V2 and V3:

(0, 2)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Data:L8a4/V 2,1 Data:L8a4/V 3,1 Data:L8a4/V 4,1 Data:L8a4/V 4,2 Data:L8a4/V 4,3 Data:L8a4/V 5,1 Data:L8a4/V 5,2 Data:L8a4/V 5,3 Data:L8a4/V 5,4 Data:L8a4/V 6,1 Data:L8a4/V 6,2 Data:L8a4/V 6,3 Data:L8a4/V 6,4 Data:L8a4/V 6,5 Data:L8a4/V 6,6 Data:L8a4/V 6,7 Data:L8a4/V 6,8 Data:L8a4/V 6,9

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$-1 is the signature of L8a4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 χ 6                 1 -1 4               2   2 2             2 1   -1 0           4 2     2 -2         3 4       1 -4       3 2         1 -6     1 3           2 -8   1 3             -2 -10   1               1 -12 1                 -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Link[8, Alternating, 4]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 4]]
Out[3]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 10, 13, 9], X[16, 13, 5, 14], X[14, 7, 15, 8], X[8, 15, 9, 16], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[4]:=	GaussCode[Link[8, Alternating, 4]]
Out[4]=	GaussCode[{1, -7, 2, -8}, {7, -1, 5, -6, 3, -2, 8, -3, 4, -5, 6, -4}]
In[5]:=	BR[Link[8, Alternating, 4]]
Out[5]=	BR[Link[8, Alternating, 4]]
In[6]:=	alex = Alexander[Link[8, Alternating, 4]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 4]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 4]], KnotSignature[Link[8, Alternating, 4]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[8, Alternating, 4]][q]
Out[10]=	-(11/2) 2 4 6 5 6 3/2 q - ---- + ---- - ---- + ---- - ------- + 4 Sqrt[q] - 3 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[8, Alternating, 4]][q]
Out[12]=	-18 -16 2 -10 2 2 4 -2 6 8 3 - q - q - --- + q + -- + -- + -- + q + q - q 12 8 6 4 q q q q
In[13]:=	Kauffman[Link[8, Alternating, 4]][a, z]
Out[13]=	3 5 2 4 6 2 a 3 a a 2 z 3 5 -3 a - 3 a - a + --- + ---- + -- - --- - 7 a z - 7 a z - 2 a z + z z z a 2 3 2 z 2 2 4 2 6 2 5 z 3 3 3 z + -- + 2 a z + 4 a z + 2 a z + ---- + 11 a z + 9 a z + 2 a a 4 5 5 3 4 z 2 4 6 4 3 z 5 3 5 3 a z + 3 z - -- + 5 a z - a z - ---- - 5 a z - 4 a z - 2 a a 5 5 6 2 6 4 6 7 3 7 2 a z - 3 z - 5 a z - 2 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[8, Alternating, 4]], Vassiliev[3][Link[8, Alternating, 4]]}
Out[14]=	{0, 2}
In[15]:=	Kh[Link[8, Alternating, 4]][q, t]
Out[15]=	4 1 1 1 3 1 3 3 4 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 q q t q t q t q t q t q t q t 2 3 2 2 2 4 2 6 3 ---- + ---- + 2 t + 2 q t + q t + 2 q t + q t 4 2 q t q t